Inequalities in absolute value of real numbers.

In this post we are going to discuss about the inequalities in absolute value of real numbers which we come across while solving the questions related to inequalities.

 Absolute value of real number \(\;x\;\) also known as numerical value or modulus of real number is denoted by \(\;|x|\;\) is defined as

$$|x|=\begin{cases}x \;&\;\text{if }\;x\geq 0\\-x;& \;\text{if}\;x\lt 0\end{cases}$$

in a fun way we can say the \(\;|x|\;\) is a  person with positivity, which take negativity and convert into positivity, for example \(\;|-2| =  2 \;\), and \(\; |2| = 2\;\) , so this function only act on negative values , but keeps  positive values as it is.

After this we are going to discuss some inequalities properties used in absolute value for real numbers .( Here we are specific about real numbers because for complex numbers these properties may vary)

 if  \(\; a, b \in \mathbb{R}\;\) then

1)   \(\displaystyle  |a|^{2} = |-a|^2=a^2\)

2)  \(\displaystyle  |ab|=|a||b|\)

3)  \(\displaystyle \left |\frac{a}{b} \right |= \frac{|a|}{|b|}\) provided \(\; b \neq 0\)

4)  \(\displaystyle |a+b| \leq |a| +|b|\;\;\) also known as Triangle Inequality

5)  \(\displaystyle |a-b| \geq  \bigg| |a|-|b| \bigg | \;\;\) also known as Triangle Inequality

6)  \(\displaystyle |a-b| \leq |a|+|b|\)   

7)  \(\displaystyle |a+b| \geq \bigg| |a|-|b|\bigg |\)

8)  \(\displaystyle |a +b| = |a| +|b| \;\;\text{if and only if }\; ab \geq 0\) 

9)  \(\displaystyle  |a +b| < |a| +|b| \;\;\text{if and only if }\; \;ab < 0\) 

10)  \(\displaystyle \sqrt{a^2+b^2} \leq |a| +|b|\)

11)   \(\displaystyle \sqrt{|a+b|} \leq \sqrt{|a|}+\sqrt{|b|}\)

12)  \(\displaystyle \frac{|a+b|}{1+|a+b|}  \leq \frac{|a|}{1+|a|} +\frac{|b|}{1+|b|}\)

13) \(\displaystyle |x| \leq a\; \Rightarrow \;-a \leq x \leq a\;\;\)\( \text{given a > 0}\)